The phrase “next in turn” means induction. It has the same meaning of following the chain and each does the activity and thus makes something true of that link or stage.
Next in turn to go to the bank teller, doctor, grocery check out means it becomes true that you have done your transaction.
If the first person in line checks into the plane, and each person in turn does, then eventually all in line have checked in.
This is math induction. It is something a first grader can understand.
Math induction is not limited to infinite sets. Math induction applies to finite sets and should be taught in terms of them. Sets with 1, 2 and 3 members are a good start. This will make it seem an ordinary idea.
Mathematical induction is no harder than standing in line to wait your turn.